Cobordisms of graphs: A sliceness criterion for stably odd free knots and related results on cobordisms

In [1], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper, we give a similar criterion for stably odd free knots. In essence, free knots may be considered as framed 4-graphs. That leads to an important notion of framed 4-graph cobordism and the associated notion of graph genera. Some additional results on graph and free knot sliceness and cobordism are given.